Clique Problem

The clique problem, focusing on finding complete subgraphs within larger graphs, is a fundamental challenge in computer science with applications ranging from social network analysis to robotics. Current research emphasizes algorithmic improvements, particularly through branching algorithms, semidefinite programming relaxations, and evolutionary approaches, to solve variations like the maximum k-defective clique and densest biclique problems, often incorporating techniques like branch-and-cut and cutting-plane methods. These advancements aim to improve efficiency and scalability for increasingly large and complex graph instances, impacting fields requiring efficient subgraph identification and optimization. Furthermore, connections between clique problems and other graph-related challenges, such as biclustering and graph coloring, are actively being explored, leading to the development of novel algorithms and improved understanding of their computational complexity.